Lemoine2nd

1. Second Lemoine circle

Given triangle ABC, consider the antiparallels DI, EG, FH to its sides passing through its symmedian point K.
[1] The intersection points of these lines with the sides of the triangle define three diameters of a circle whichis called second Lemoine circle of the triangle and is centered at the symmedian point K.
[2] Hexagon DQEIFG is symmetric with sides parallel to those of the triangle and non-maximal diagonals parallel to the altitudes of the triangle.
[3] Lines LO, PN, JM, joining the middles of the sides of ABC with the middles of corresponding altitudespass through the symmedian point.

Quadrangle ABQF is cyclic, hence angle(KQE) = A. Similarly prove angle(KEB) = A, hence KQE is isosceles and analogously GKD, FKI are isosceli. K being the middle of QF, GE, DI completes easily the proof of [1,2]. Notice that triangles GIQ, DEF are antipodal (hence congruent) w.r. to the Lemoine circle and have their sides orthogonal to corresponding sides of ABC, hence they are similar to it.
[3] follows from an easy calculation in trilinears. In fact the trilinears of L, O, K are respectively.
(0, c, b), (1, cos(C), cos(B)), (a, b, c).
The obvious identity for the determinant:

Implies that line OL contains K.
Another, more elementary proof is given below.
A third proof making some interesting remarks is contained in [3] see Honsberger p. 65.
Notice that K, is X(69) with respect to JLP with corresponding trilinears (cos(A)/a², ... ).

2. Rectangles in triangles

Inscribe rectangles into triangles by the following procedure singling out side BC.
1) Start with a point Q of BC and draw QQ' orthogonal to BC, Q' being on AB.
2) From Q' draw orthogonal Q'Q'' to QQ', Q'' being on AC.
3) Draw Q''Q''' orthogonal to Q'Q'', Q''' being on BC.
The middle P of the resulting rectangle QQ'Q''Q''' which is inscribed in the triangle moves on aline passing through the middles {F, E} of side BC and altitude AD, as Q varies on side BC.

Proof by picture.
This simple exercise implies an easy proof of [3] of the previous section. In fact, in the course of proof of properties [1] and [2] it was seen that the symmedian point K is the center of three rectangles inscribed inthe triangle in the previous sense. Thus K is on the intersection of the three lines like EF resulting from the corresponding loci with respect to the three sides of the triangle.

References

Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington DC, Math. Assoc. Ammer., 1995, pp. 65.

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1. Second Lemoine circle

2. Rectangles in triangles

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References